A Global Spline Space with Improved Geometry Consistency Near Extraordinary Vertices
Graphical abstract
Introduction
For defining complex engineering models, one often uses an unstructured quadrilateral control mesh having extraordinary control vertices with the number of incident edges . Smooth tensor product splines can be easily defined over a regular mesh with structured quadrilateral elements. However, there is no canonical construction method of smooth splines over unstructured meshes with extraordinary control vertices. In the literature, one can find two general classes of methods to deal with unstructured quadrilateral meshes, namely subdivision methods and patch work methods. Subdivision methods are efficient to use, but they result in non-polynomial extraordinary surface patches and the continuity at extraordinary positions is often restricted to or with bounded curvature. Patch work methods can also be used with explicit Bézier representations and to meet higher order continuity requirements, mostly up to continuity at extraordinary positions.
Catmull–Clark subdivision [1] and Loop subdivision [2] are the most popular subdivision schemes in animation industry which produce continuous surface, over a control mesh of arbitrary topology consisting of quadrilateral and triangle faces, respectively. A set of smooth spline basis can be defined using subdivision approach, a scaled spline ring will be added to the hole around extraordinary vertex after each step of refinement. This would result in infinite piecewise representations near an extraordinary vertex, and Stam [3] provided a method to compute the basis function and corresponding derivatives at any point, but there is no accurate quadrature technique developed for the subdivision surfaces.
Reif [4] presented sufficient conditions for subdivision surfaces to be tangent plane continuous at positions of extraordinary vertices, namely extraordinary positions, if the respective subdivision matrix has two identical subdominant eigenvalues and if the characteristic map defined by the two corresponding eigenvectors is regular and injective. Reif [5] concluded that a stationary curvature continuous polynomial scheme requires at least degree 6, to produce limit surfaces at extraordinary positions. Prautzsch and Umlauf [6] further constructed a -subdivision scheme allowing for negative weights with a larger subdivision mask, and the subdivision surface will have a flat point at extraordinary positions with zero curvature. Peters and Reif [7], [8] and Reif [9] stated the necessary and sufficient conditions for a subdivision surface with generic input data to be curvature continuous at an extraordinary vertex. Karčiauskas and Peters [10] developed a guided subdivision algorithm through sampling from a guide surface after reparameterization and can achieve continuity with degree bi-. Karčiauskas and Peters [11] also presented a algorithm of polynomial degree bi-6 and a curvature bounded algorithm of polynomial degree bi-5. They further reported in [12] another algorithm of polynomial degree bi-5 emphasizing rapid subdivision through adjusting the contraction speed. However, these subdivision schemes with curvature continuity at extraordinary positions would increase the complexity and computation cost for practical applications.
Tuned subdivision schemes have also been widely investigated by optimizing the limit properties of a subdivision surface and are comparatively easy to implement. Sabin [13] tuned subdivision stencils of one-ring newly inserted face and edge vertices together with new extraordinary vertices to satisfy quadratic equality constraints on subdominant eigenvalues. Sabin et al. [14] constructed stencils using the second divided differences and obtained the same stencils for one-ring new vertices as that of [13], except that for the extraordinary vertex. Thus they produce the same curvature behavior since both of the schemes produce the same quadratic maps. Loop [15] developed a tuning subdivision scheme based on Loop subdivision [2] with bounded curvature and convex hull property. However one can observe a flat area with zero local curvature around extraordinary positions. Barthe and Kobbelt [16] demonstrated a tuning framework for subdivision rules near an extraordinary vertex and optimized the eigenvectors of the subdivision matrix to approximate necessary smoothness conditions and force subdominant eigenvalue of the subdivision matrix approaching to 0.5. Augsdörfer et al. [17] minimized the Gaussian curvature variation near an extraordinary vertex of a representative sequences of central surfaces for tuning subdivision masks of several bounded curvature subdivision schemes. The optimal large subdominant eigenvalue would lead to severe polar artifact and reduced convergence rate. Cashman et al. [18] proposed constant multipliers modifying the weights of extraordinary vertices contributing to different new vertices to satisfy bounded curvature constraints, and a simplified optimization with respect to the distribution of natural configuration is performed to determined the parameters. The same multipliers are used for the refine and smooth stages, which leads to complications for deriving the subdivision stencils.
Ginkel and Umlauf [19] also provided an energy optimization framework based on the parameterization dependent energy functional for each step of subdivision. The subdivision scheme is thus not stationary and dependent on the control mesh and bounded curvature is not ensured. Zhang et al. [20] optimized the subdivision stencils for valence through minimizing a thin-plate energy function and approximate the second order eigenbasis to specific quadratics of the characteristic map. They demonstrated two sets of optimized subdivision stencils for valence with respect to cup-like and saddle-like input mesh, respectively. Li et al. [21] proposed a hybrid non-uniform subdivision scheme with non-negative knot intervals near an extraordinary vertex and with improved convergence rate for isogeometric analysis. Primal and dual subdivision schemes were combined to define subdivision rules with subdominant eigenvalue through manipulating the eigen structure. Ma and Ma [22] presented a bounded curvature subdivision space and optimized the second order eigenbasis functions towards quadratic functions of the characteristic map to obtain the best possible curvature performance. The curvature variation of the limit surface is better or comparable to that of Augsdörfer et al. [17]. They further investigated in [23] bounded curvature subdivision stencils with the least polar artifact while maintaining the best possible bounded curvature performance based on the work in [22]. Uniform refined mesh and visually smooth limit surface can be obtained for valences and to .
One can also construct finite smooth patches near an extraordinary vertex, through introducing geometry continuity constraints for adjacent patches. Existing cap construction methods in an extraordinary region can produce a surface mainly based on a Catmull–Clark control mesh, with different degrees, connecting maps and optimization functionals applied. Prautzsch [24] and Reif [25] showed how to construct finite bisextic patches filling an n-sided hole, but such an extraordinary patch involves rings of spline control vertices. Loop [26] and Loop and Schaefer [27] filled the n-side hole near an extraordinary vertex of Catmull–Clark surface with finite patches of bi-degree 7 and proposed an energy functional minimizing the 4th order derivatives. Karčiauskas and Peters [28] explored the space of caps using 2 × 2 degree bi-5 patches and tested various functionals to eliminate degrees of freedom to improve the curvature behavior near an extraordinary vertex. They also constructed in [29] completion of bicubic spline surfaces with minimal degree 6 using rational connecting map between sectors and 5th order derivative functional to solve the undetermined system.
There are also cap construction methods designed for engineering analysis with limited continuity at extraordinary position. Karčiauskas et al. in [30] constructed bi-4 spline elements near an extraordinary vertex with sub-optimal convergence rate for Poisson’s equation on the disk. Scott et al. [31] reported unstructured T-splines with elevated degree patches that ensure continuity near extraordinary positions, but exhibit sub-optimal approximation property for IGA. Recent works based on a D-patch framework developed by Reif [32] demonstrate excellent approximation property. Nguyen et al. [33] constructed spline elements near irregular points combined with PHT splines based on the D-patch construction. The constructed spline spaces are refinable and demonstrate almost-optimal convergence in and norms for a Poisson problem. Toshniwal et al. [34] constructed a set of spline basis functions that can produce optimal or slightly sub-optimal convergence rate, which is due to the extra degrees of freedom introduced by local face-based control points through refinements. The condition number is also high for isogeometric analysis, especially for refined meshes. Wei et al. [35] developed blended B-spline construction for unstructured quad/hex meshes with continuity near an extraordinary vertex but with optimal convergence rates. This method introduces less degrees of freedoms comparing to that of Toshniwal et al. [34]. However, most of the above construction methods designed for engineering analysis produce surfaces with continuity at most and are often complicated to implement compared with traditional subdivision schemes.
Section snippets
A brief overview of the proposed spline space
This article presents a spline space that fills in 2-ring extraordinary surface patches with curvature continuous bi-7 degree Bézier surface patches and with the least geometry consistence error among different levels of refinement. The spline space is built upon a bounded curvature subdivision scheme for quadrilateral meshes with the least polar artifacts (LPA). The LPA subdivision scheme is adapted from that reported in [23] and the optimized subdivision stencils are produced with
-spline constraints for exact satisfaction
In this article, we use basis labeling as shown in Fig. 2. Individual Bézier control points are labeled as shown in Fig. 3.
Local shape energy minimization
We first introduce the shape energy functional for the two-ring elements of the th sector, which has a similar formulation as that used in [27] where is the target tensor product patch and this functional would approximate the target patches to a bicubic patch.
Eq. (14) can be further written as where is a 256 × 256 matrix, with and denotes the 1 × 64 vector of
Results and further discussions
For simplicity, we use G2-LPA representing basis functions constructed by optimizing shape energy only in Section 4.1 subject to constraints and support constraints using LPA subdivision (), and G2-LPA-GC representing basis functions constructed by optimizing the hybrid objective function inSection 4.3 in terms of shape energy and geometry consistency error energy subject to constraints, geometry sampling constraints and support constraints, using LPA subdivision (). For
Conclusions
In this paper, we present a novel global spline space which fills in extraordinary surface patches of a subdivision scheme with curvature continuous bi-7 Bézier patches. The scheme is constructed based on a bounded curvature subdivision scheme with least polar artifact requiring that the subdominant eigenvalue , and produces two-ring extraordinary surface patches near the extraordinary vertex. The two-ring patches are required to satisfy necessary continuity constraints with outer
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work described in this paper was supported by GRF Research Grants (Project Nos. CityU11206917 and CityU11201919) from the Research Grants Council of the Hong Kong Special Administrative Region, China .
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